Composition factors of Specht modules for Hecke algebras of type An

We examine the composition factors of Specht modules for Hecke algebras of type An at roots of zero, and their positions in the Jantzen–Schaper filtration. Each Specht module is decomposed into a direct sum of orthogonal subspaces corresponding to residue classes of standard tableaux; similarly for the Gram matrix. We show that, for a given subset of these classes, the corresponding invariant factors of the Gram matrix over a local ring completely determine the decomposition matrix and Jantzen–Schaper filtration.From this we deduce elementary proofs for a number of well-known results, notably the James “first column” theorem together with its generalisation by Donkin, and the determination of the decomposition matrices for two-part partitions for the symmetric group algebra and for general linear groups. We extend these to analogous results for the Jantzen–Schaper filtration; in particular, we derive a closed formula for the Jantzen indices for two-part partititions (two-column diagrams in our formulation) in the case of the Hecke algebra.